Quantum information approach to normal representation of extensive games
We modify the concept of quantum strategic game to make it useful
for extensive form games. We prove that our modification allows to
consider the normal representation of any finite extensive game
using the fundamental concepts of quantum information. The
Selten’s Horse game and the general form of two-stage extensive
game with perfect information are studied to illustrate a
potential application of our idea. In both examples we use
Eisert-Wilkens-Lewenstein approach as well as Marinatto-Weber
approach to quantization of games.
Keywords Extensive game Normal representation Quantum game Nash equilibrium
Mathematics Subject Classification (2000) 81P68 91A18 91A80
Over the period of twelve years of research on quantum games , the idea of quantum strategic game  has been well established. From mathematical point of view, the quantum information approach to a game is described by the four-tuple:
The Hilbert space is the place of the game and the sets of unitary operators play the role of strategy sets for the first and the second player, respectively. Given a unit vector (the initial state), the players each choose a unitary operator changing the vector into the vector (the final state). The last components are the functionals for . They imitate payoff functions for the players assigning a real number to the final state . It turns out that the four tuple (1) generalizes playing a classical game. The two well-known ways based on the framework (1): the Eisert-Wilkens-Lewenstein (EWL) scheme  and the Marinatto-Weber (MW) scheme  show that a game can be successfully written in the form (1): it is possible to set each of the four components, so that a game and the corresponding game defined by the four-tuple (1) are the same with respect to a game-theoretic analysis. Another key feature is that the protocol (1) allows to achieve results unavailable in the game played classically (see, for example,  and ). In our paper we are going to deal with extensive games in quantum domain. In spite of quite many researches connected with this issue (for instance, concerning quantum Stackelberg duopoly  and ), there is no generally accepted framework for playing quantum extensive games by now. Interestingly, we have shown in  that (1) may indeed be useful for extensive games with imperfect information. In this paper, we extend our previous idea. We prove that the slight modification of allows to obtain normal representation of extensive games in the quantum domain.
2 Preliminaries to game theory
Definitions in the preliminaries are based on . This section starts with a definition of a finite extensive game (without chance moves).
Let the following components be given:
A finite set of players.
A finite set of finite sequences that satisfies the following two properties:
the empty sequence is a member of ;
if and then .
Each member of is a history and each component of a history is an action taken by a player. A history is terminal if there is no such that . The set of actions available after the nonterminal history is denoted and the set of terminal histories is denoted .
The player function that points to a player who takes an action after the history .
For each player a partition of with the property that for each and for each , an equality is fulfilled. Every information set of the partition corresponds to the state of player’s knowledge. When the player makes move after certain history belonging to , she knows that the course of events of the game takes the form of one of histories being part of this information set. She does not know, however, if it is the history or the other history from .
For each player a utility function which assigns a number (payoff) to each of the terminal histories.
A five-tuple is called a finite extensive game.
Our deliberations focus on games with perfect recall (although Def. 2.1 defines extensive games with imperfect recall as well) - this means games in which at each stage every player remembers all the information about a course of the game that she knew earlier (see  and  to learn about formal description of this feature).
The notions: action and strategy mean the same in static games, because players choose their actions once and simultaneously. In the majority of extensive games a player can make her decision about an action depending on all the actions taken previously by herself and also by all the other players. In other words, players can make some plans of actions at their disposal such that these plans point out to a specific action depending on the course of a game. Such a plan is defined as a strategy in an extensive game.
A pure strategy of a player in a game is a function that assigns an action in to each information set .
Like in the theory of strategic games, a mixed strategy of a player in an extensive game is a probability distribution over the set of player ’s pure strategies. Therefore, pure strategies are of course special cases of mixed strategies and from this place whenever we shall write strategy without specifying that it is either pure or mixed, this term will cover both cases. Let us define an outcome of a pure strategy profile in an extensive game without chance moves to be a terminal history that results if each player follows the plan of . More formally, is the history such that for we have .
Let an extensive game be given. The normal representation of is a strategic game in which for each player :
is the set of pure strategies of a player in ;
defined as for every and .
One of the most important notions in game theory is a notion of an equilibrium introduced by John Nash in . A Nash equilibrium is a profile of strategies where the strategy of each player is optimal if the choice of its opponents is fixed. In other words, in the equilibrium none of the players has any reason to unilaterally deviate from an equilibrium strategy. A precise formulation is as follows:
Let be a game in strategic form. A profile of strategies is a Nash equilibrium if for each player and for all :
A Nash equilibrium in an extensive game with perfect recall is a Nash equilibrium of its normal representation, hence Def. 2.4 applies to strategic games as well as to extensive ones.
3 Preliminaries to quantum computing
In this section we give a brief overview of the Dirac notation and basic terms of quantum information. The preliminaries are based on  and are sufficient to study the paper. Nonetheless, we encourage the reader unfamiliar with techniques from theory of quantum information to consult  and, for example, .
First of all we adopt the convention that instead of denoting vectors by boldface letters, e.g. , they are denoted as kets: .
Let be a vector space with the fixed basis and let
The vector can be also written in the column matrix notation
Let be now regarded as a Hilbert space and . The inner product of the vector with the vector will be denoted by . The notation is used for the dual vector to . The dual vector (also called bra) is a linear operator defined by Thus, the inner product requirements imply that
The common assumption in quantum computing is to consider Hilbert space with an orthonormal basis. Let us denote the basis as (also called computational basis). Let and be the vectors with respect to the basis . Then the inner product can be expressed in terms of matrix multiplication:
In this case, the dual vector has a row matrix representation whose entries are complex conjugates of the corresponding entries of the column matrix representation of .
The fundamental concept of quantum information is quantum bit (qubit) described mathematically as a unit vector in a Hilbert space . According to the notation explained above:
The measurement of a qubit with respect to an orthonormal basis (not necessarily in the computational basis) yields the result or with probability leaving the qubit in the corresponding state or . In particular, measuring the qubit given by (7) with respect to results in the outcome 0 with probability and the outcome 1 with probability , with post-measurement states and , respectively.
Suppose and are Hilbert spaces with orthonormal bases and , respectively. Then the tensor product is a Hilbert space of dimensionality with the orthonormal basis . The matrix representation of an element is the Kronecker product of respective matrix representations of and . In the further part of the paper we use the abbreviated notation or for the tensor product .
A system of qubits is described as a unit vector in the tensor product space that has -element computational basis
Thus, it is described by the vector
We say that the state (9) is separable if it can be written as for some , . The dual vector is defined in the same way as in (5). Similarly, the measurement of the state given by (9) with respect to an orthonormal basis yields the result with probability . Otherwise, the state is called entangled.
We use the Dirac notation throughout the whole paper. However, each of the results below can be easily reconstructed using the matrix notation.
4 Normal representation of extensive games in quantum domain
From that moment on, we will consider extensive games with two available actions at each information set so that we could use only qubits for convienience. Any game richer in actions can be transferred to quantum domain by using quantum objects of higher dimensionality.
Let us extend the protocol (1) to include components making it useful for extensive games. Such a quantum game is specified by a six-tuple:
where the components are defined as follows:
is a complex Hilbert space with an orthonormal basis .
is a set of players with the property that .
is the initial state of a quantum system of qubits .
is a surjective mapping. A value indicates a player who carries out a unitary operation on a qubit .
For each the set is a subset of unitary operators from that are available for a qubit . A (pure) strategy of a player is a map that assigns a unitary operation to a qubit for every . The final state when the players have performed their strategies on corresponding qubits is defined as:
For each the map is a utility (payoff) functional that specifies a utility for the player . The functional is defined by the formula:
There are only two additional components in (10): and , in comparison with (1). They completely specify qubits to which a player is permitted to apply her unitary operator. Notice also that the protocol of quantization of strategic games according to  is obtained from by putting . We claim that such addition together with appropriate fixed values in (12) are sufficient for considering an extensive game in quantum domain (of course, if the assumption that the tuple correctly describes strategic games in quantum domain is true). The line of thought is as follows. Any strategic game can be considered as a special case of an extensive game where players move sequentially but each of them does not have any knowledge about actions taken by the other players. In other words, each player in a strategic game has exactly one information set in which she takes an action. Thus, in a simple case of bimatrix game, the scheme (1), in fact, identifies an operation on a qubit with player’s move made at her unique information set, and then the individual game outcomes are assigned to appropriate measurement results. An extensive game can have many information sets, and more than one of them can be assigned to the same player. Therefore, our extension of is aimed at similar identification for extensive games. As a result, we obtain that we are able to write in (10) the normal representation of an extensive game.
Before we formulate the formal statement, notice first that the tuple (10), in fact, determines some game in strategic form in the sense of classical game theory. If and are fixed, each player chooses her strategy from a set and then the associated utility is determined. Therefore, it always makes sense to associate (10) with some . Secondly, let us specify a sufficient condition of equivalence for two strategic games and . Namely, if there is a bijective mapping for each such that for each profile we have where , then the games are isomorphic (to find out more about isomorphisms of strategic games see ). Now, we can formulate the following proposition:
Let be a finite extensive game with two available actions at each information set. Then there exists a six-tuple (10) that specifies a game isomorphic to the normal representation of .
Proof. Let us consider an -player extensive game with information sets. In addition, let us assume two-element set of available actions in each information set . We specify components of the tuple as follows. Let be the computational basis of and let the initial state be of the form , where is some fixed state of . Let us restrict the set of available operators on to the set of two operators where is the identity operator and is the bit-flip Pauli operator. This specification implies that for any mapping specified in (10) each strategy profile is an operator of the form , where . Thus, for each strategy profile there is some such that
Let us fix a bijective mapping between and the set of all players’ information sets of . Since for each player and history we have we can simply take . Then the correspondence associates each information set of each player with exactly one qubit. As and a number of information sets of is equal to a number of qubits, a set of strategies of the normal representation of and a strategy set of quantum game defined by the tuple (10) are equinumerous (with cardinality equal each) for each . Therefore, for each , we can define a bijective mapping . These mappings induce the following bijection between the sets of strategy profiles:
The equations in (13) imply that for all we can
select numbers in (12) in a
way that where is the utility
function of the normal representation of . Such
specification of (10) makes it isomorphic to the
normal representation of .
Many researches on quantum games played via scheme (1) are based on appropriately fixed basis for a space, the initial state, and a range of available unitary operators, in order to obtain interesting properties of a quantum game. We will use the two best-known configurations of : the Marinatto-Weber (MW) scheme  and the Eisert-Wilkens-Lewenstein (EWL) scheme  to examine extensive games via the protocol (10) (see also  and  for other applications of these schemes). In the former scheme players are allowed to use only the identity operator and the bit-flip Pauli operator. The results superior to classical results are obtained by manipulating the initial state . The later scheme allows to use broader range of unitary operators (including also the whole set ). The following examples concern both settings.
To convert the following games into quantum ones, we use the same reasoning as in the proof of Proposition 4.1. The first example deals with a case where each player operates on one qubit.
Let us consider a three player extensive game:
determined by the following components:
, , .
The game is depicted in Fig. 1.
It is the Selten’s Horse game  with modified
payoffs. Since each of the players has one information set, their
sets of strategies are , , and
, respectively. Profiles: and
are the only pure Nash equilibria in this game
and indeed each of them could be equally likely chosen as a
scenario of the game. The utilities for players 1 and 2 assigned
are higher than the utilities corresponding to - a desirable profile for player 3. The
uncertainty of a result of the game follows from the peculiar
strategic position of player 3. She could try to affect the
decision of others by announcing before the game starts, that she
is going to take an action . If the statement of player 3
is credible enough then the history might occur.
The MW approach. Let us examine the Selten’s Horse game via the protocol (10). It turns out that among quantum realizations of the game there exist ones that provide the players with a unique reasonable solution. One of these realizations is constructed, according to the idea of the MW scheme, as follows:
is a Hilbert space with the computational basis , where for ;
the initial state takes the form:
is an identity mapping defined on ;
the payoff functionals are defined as follows:
Let us first determine the utilities associated with any profile , where for . The final state after the operation takes the form:
where in the negation of . Using the last equation and formula (12) the expected utilities, for example, for become:
All possible values are shown in Fig. 2.
Now, we can analyze the game like a classical strategic game. To be fully precise, such equivalence is assured since we can extend to use mixed strategies. A profile of mixed strategies in the game in Fig. 2 determines a probability distribution over profiles . Thus, the mixed strategy outcome in (16) is described simply as an ensemble . Then both the ensemble and an appropriate profile of mixed strategies in the game in Fig. 2 generate the same utility outcome.
Let us notice now that is a generalization of the normal representation of . We are able to get the normal representation of out of Fig. 2 putting the initial state , i.e., by putting into the matrix representation of the game in Fig. 2. Moreover, the same is the case for as well as any initial state being a basis vector of . Then the game coincides with the classical Selten’s Horse game up to the order of players’ strategies.
Now, we examine the six-tuple (16) to find a reasonable solution for players. Let us determine pure Nash equilibria in the game by solving for each profile , where the system of inequalities imposed by the condition (2). Using values of placed in Fig. 2, we find that, for example, the profile constitutes the Nash equilibrium if and only if . Further investigation shows that the profile also fulfills (2) with the requirement and that there are no other pure Nash equilibria. Taking into consideration we conclude that
Let us assume results of the games and to be an equilibrium in pure strategies. Then formula (21) shows that each player can gain from playing game . In classical case players 1 and 2 can assure themselves 2 utility units and player 3 can get 1 unit for sure by playing pure equilibria. All these payoffs are strictly less than the payoffs corresponding to pure Nash equilibria in , irrespectively of what is a value of . Moreover, notice that there is the unique equilibrium in the game if and the same utilities are assigned to both equilibria in the case . This implies that in the game the strategy profile for and the strategy profile for are reasonable profiles for all players.
An interesting fact worth pointing out is that for
arbitrary close to 0 or (i.e. for angles defining
the classical game) the equilibrium is unique. This discontinuity
implies possible applications of quantum games to classical game
theory. Namely, the MW approach may serve as a Nash equilibrium
refinement by considering only Nash equilibria that hold out some
slight perturbation of . In fact,
the profile is the unique
pure trembling hand perfect equilibrium in
 (see also , example 252.1). Although
a further investigation is required, we believe there is a strong
connection between the above method and the Selten’s concept of trembling hand equilibrium.
The EWL approach. The second quantum realization of is in the spirit of the EWL protocol. Contrary to the previous one, where the number of reasonable Nash equilibria was reduced to the unique one, we focus this time on improving strategic position of only one of the players. Namely, let us modify the previous quantum game as follows:
is a Hilbert space with the basis of entangled states defined as follows:
the unitary strategies are elements of whose matrix representation with respect to the computational input and output basis is the following:
the payoff functional is derived from with respect to basis states (23):
In this case only the third player is allowed to use unitary strategies beyond the set of one-parameter operators of the game by using the additional parameter . We demonstrate now that such extended strategy set of player 3 significantly improves her strategic position. In order to see this, let us determine the expected utility for each player that corresponds to a profile of strategies (since angles specify strategies of player 1, 2, and 3, we denote them, for convenience, as and , respectively). Using formula (11) the final state associated with a profile is the following:
As it should be expected in the EWL protocol, we get the classical game when the player 3 is also restricted to use only the strategies . Namely, let us put , , and to Eq. (28). Then we obtain
Formulae (29) are exactly the players expected payoffs in the classical game if they choose their actions and with probability and respectively. Thus, in the particular case, if , we obtain payoffs corresponding to pure strategy profiles in .
Now we solve a problem how player 3 can gain from using 2-parameter operators as her strategies. An interesting feature is that the game keeps the pure Nash equilibria of the game , i.e., the profiles: and - equivalents for the respective Nash equilibria profiles and in - are Nash equilibria profiles also in . However, unlike in the game , there is another pure equilibrium where for each player . This non-equivalence to the classical profile is essential for strategic position of player 3. She can force the other players to play strategies from the profile (instead of - their the most preferred equilibrium) by making an announcement that she is going to play . The other players know that this threat is credible enough as the player 3 does not suffer a loss when she deviates from to , since . However, the opponents of the third player lose 3 utilities, since . Furthermore, given fixed, rationality demands that they play strategies dictated by as we have . This argumentation allows to treat the profile as reasonable solution of . Thus, the strategic position of the player 3 has been significantly improved in comparison to her classical strategies.
The second example is aimed at showing that the proposed scheme of playing extensive games based on the six-tuple (10) can be applied to extensive games in which some of players have more than one information set. Unlike in Example 4.2 we now focus only on converting an extensive game into the form described by (10) without considering a specific strategic situation.
Let the following extensive game be given:
, ; , .
Like in the previous example, the game has three information sets in which two actions are available. However, in this case, two information sets represent the knowledge of player 2. Thus, she specifies an action at each of them. The game is illustrated in Fig. 3.